SEPARATING QUASI-CONVEX SUBGROUPS IN 7-SYSTOLIC GROUPS

FRÉDÉRIC HAGLUND AND JACEK ŚWIA̧TKOWSKI

Abstract. Let Γ be a group acting without inversions and simply transitively on the top-dimensional simplices of some simply-connected simplicial complex X with “simplicial negative curvature”. Then the quasi-convex subgroups of Γ are convex-cocompact. Furthermore, if the action of Γ on X satisfies some additional condition called ”extra-tilability”, the quasi-convex subgroups of Γ are separable, i.e. every such subgroup is the intersection of finite index subgroups. The latter result applies to a large class of ”simplicially negatively curved” groups constructed in [JS2] and [Hag2].

Contents

1. Introduction 1 2. Systolic complexes: definitions and basic results 4 2.1. Local k-largeness, k-systolicity and consequences 4 2.2. Convexity and saliency 5 2.3. Balls and spheres 6 3. Extra-tilability and local extra-tilability for group actions. 7 4. Systolic complexes: further results. 10 4.1. Ladders and bigons 10 4.2. Geodesic convexity 12 5. Quasi-convexity in 7-systolic complexes 13 6. QCS for 7-systolic simplices of groups 16 References 19

1. Introduction

Separability properties attracted attention of researchers since the emergence of combinatorial group theory. The simplest such property is residual finiteness. One of its statements says that a group is residually finite if the intersection of its all finite index subgroups is the trivial subgroup {1}. A more general concept is that of separability of a subgroup. A subgroup H < G is separable if it can be expressed as intersection of certain family of finite index subgroups of G. For example, residual finiteness of G is equivalent to separability of the trivial subgroup {1} < G.

Date: June 26, 2006.

Research supported by grants from NSERC and FCAR and the Polish Scientific Research Committee (KBN) grant 2 P03A 017 25.

Residual finiteness of a group G allows solution of the word problem in G and implies that G is hopfian (i.e. every surjective homomorphism of G to itself is an automorphism). We refer the reader to [Mag] for a survey of those and other early developments concerning residual finiteness. Separability of a subgroup H < G implies that the membership problem for H is decidable [BN]. The geometric interpretation of separability is that it is possible to remove self-intersections of π_{1}-injective immersions (with separable image) by lifting them to finite sheeted covers. This has been used mainly in low dimensional topology, see [Sc].

Separability properties are difficult to establish or disprove. For example, it is not known whether there exists a word-hyperbolic group which is not residually finite. The property that every finitely generated subgroup is separable (so called subgroup separability) is established for free groups [Hal], surface groups [Sc], finitely generated nilpotent groups [Mal] and few other classes of groups (see [G1], [G2], [MR], [Wi2] and the references cited there).

A few years ago D. Wise suggested and initiated the study of the property (denoted QCS below) of separability for all quasi-convex subgroups of a group, rather than for all finitely generated subgroups. The quasi-convexity of a subgroup, in general, depends on a choice of a metric space on which the group acts geometrically (e.g. a Cayley graph). However, in case of word-hyperbolic groups (which we will be interested in) the property is intrinsic. We recall this notion in Definitions 5.4 and 5.6.

Hsu and Wise indicated in [HsW] examples of groups that are QCS and not subgroup separable. Wise showed the importance of the question about QCS for the problem of residual finiteness of word-hyperbolic groups ([KW], [Wi3, 11.8]). The question about QCS seems to be well posed for CAT(0) groups, since their quasiconvex subgroups are known to have decidable membership problem (and the same is true for separable subgroups). QCS is established for graph products of finite groups [Hag3], for certain class of Artin groups called tree groups [HsW], for Bianchi groups [ALR], and for certain vast class of groups that includes acute angled n-gons of finite groups with n ≥ 4 [Wi3]. On the other hand, QCS does not hold for all CAT(0) groups since there are compact CAT(0) squared 2-complexes whose fundamental groups are not even residually finite (see [Wi1], [BM]). In [HaW] it is proved that the fundamental group of a negatively curved cube complex is QCS if and only if it has a virtual embedding as a “convex cocompact” subgroup into some right-angled Coxeter group.

In this paper we study property QCS for systolic groups. These groups were introduced recently in [JS2] (and in the unpublished preprint [Hag2]), as groups acting in a geometric way on simply connected simplicial complexes that are simplicially nonpositively curved. Simplicial nonpositive curvature is a purely combinatorial local condition (introduced in [Hag1] and [JS2]) that mimicks classical nonpositive curvature, and has similar topological and group theoretic consequences. In fact, there is an integer parameter k ≥ 6 that defines a sequence of conditions called k-systolicity.

Simplicial nonpositive curvature corresponds to 6-systolicity (abbreviated to systolicity), and every m-systolic complex or group is k-systolic when m ≥ k. It turns out that 7-systolic groups are word-hyperbolic. For every k ≥ 6 there are examples of k-systolic groups of arbitrarily large cohomological dimension. Examples of such groups, constructed in [JS2] and [Hag2], are the fundamental groups (or direct limits) of simplices of finite groups satisfying certain conditions that we call local k-largeness (related to k-systolicity) and local extra-tilability. Our inductive constructions of these groups involve their residual finiteness, so in particular the following holds.

Theorem (see Proposition 19.1 in [JS2]). Let G be a locally 6-large and locally extratilable simplex of finite groups. Then G is developable and the direct limit of G is a residually finite group.

The main result of this paper, Theorem A below, considerably strengthens the above theorem.

Theorem A (see Theorem 6.1 in the text). Let G be a locally 7-large and locally extra-tilable simplex of finite groups. Then its direct limit has separable quasi-convex subgroups.

Note that simplices of groups satisfying assumptions of Theorem A exist in arbitrary dimension, and that the resulting groups satisfying QCS occur in arbitrary cohomological dimension.

Our proof of Theorem A follows the geometric method of Scott (see also [Hag3]). It is based on an independantly interesting result, Theorem B below, concerning convex cocompactness of quasi-convex subgroups in arbitrary 7-systolic groups. In this result we use the notion of a convex subcomplex in a systolic complex, two different aspect of which were introduced and studied in [Hag1] and [JS2] respectively. (The two notions of convexity are in fact equivalent, see Proposition D below.) More precisely, we prove the following.

Theorem B (see Corollary 5.8 in the text). Let G be a 7-systolic group, i.e. a group acting properly discontinuously and cocompactly, by simplicial automorphisms, on certain 7-systolic simplicial complex X. Let H < G by any quasi-convex subgroup. Then there is a convex subcomplex Y ⊂ X which is invariant and cocompact under the action of H.

Apart from application to proving Theorem A, Theorem B (together with the fact that convex subcomplex of a k-systolic complex is k-systolic, see Lemma 2.6) implies the following.

Corollary C. For any k ≥ 7, a quasi-convex subgroup of a k-systolic group is ksystolic.

This corollary is a special case of the following result announced recently by D. Wise [Wi4]: for any k ≥ 6 any finitely presented subgroup of a k-systolic group is k-systolic. Wise’s proof is different and uses the tower argument.

In Sections 4 and 5 we prove certain new results concerning systolic geometry. These results are useful in proving our main results: Theorems A and B. However, they also seem to have generally useful nature, and we think they complement the set of basic geometric properties of systolic simplicial complexes.

Two main results of Section 4 concern geodesics. For us a geodesic in a simplicial complex X between some two vertices is a polygonal curve in the 1-skeleton of X that connects these vertices and has the smallest possible number of edges. A subcomplex Y ⊂ X is geodesically convex if it is full and any geodesic in X between any two vertices of Y is contained in Y (this is the notion of convexity considered in [Hag1]). In systolic complexes X this “global” convexity property (i.e. geodesical convexity as above) turns out to coincide with the “local” property of convexity (as considered in [JS2] and recalled in Section 2, see Definition 2.5). Namely, we prove the following.

Proposition D (see Proposition 4.9 in the text). A subcomplex Y in a systolic simplicial complex X is geodesically convex iff it is convex.

The next result (see Lemma 3.4 in [Hag1]) concerns geodesics in 7-systolic complexes. The 2-dimensional case seems to be well known in small cancellation theory.

Proposition E (see Proposition 4.7 in the text). Let (v_{0}, v_{1}, …, v_{n−1}, v_{n}) and (v_{0}, v′_{1}, …, v′_{n−1}, v_{n}) be two geodesics in a 7-systolic simplicial complex X between vertices v_{0}, v_{n}. Then for each 1 ≤ i ≤ n − 1 the vertices v_{i} and v′_{i} either coincide or are connected with an edge of X.

The main geometric result of this paper is proved in Section 5, and concerns convex neighbourhoods of quasi-convex subsets in 7-systolic complexes. For any subcomplex Y of a simplicial complex X, let B(Y, X) denote the union of all simplices of X meeting Y. For any integer n ≥ 1 define inductively the neighbourhoods B_{n}(Y, X) by B_{1}(Y, X) = B(Y, X) and B_{n+1}(Y, X) = B(B_{n}(Y, X), X).

Theorem F (see Theorem 5.5 in the text). Let Y be a quasi-convex subcomplex of a 7-systolic complex X. Then there is a natural number N such that for any n ≥ N the neighbourhood B_{n}(Y, X) is a convex subcomplex of X.

Theorem F answers the last question of [Hag1] and immediately implies Theorem B.

2. Systolic complexes: definitions and basic results

In this section we recall definitions, basic properties and some more advanced results concerning systolic complexes and groups. All simplicial complexes X in this paper are assumed to be locally finite dimensional in the following sense: for every vertex v of X there is a bound on the dimension of simplices in X that contain v. Clearly, this class contains all finite dimensional and all locally finite simplicial complexes, which are most interesting for us.

2.1. Local k-largeness, k-systolicity and consequences. Systolicity is a purely combinatorial condition for simplicial complexes (and their automorphism groups) that resembles nonpositive curvature. Main references for this subject are [Hag1] and [JS2].

A simplicial complex X is flag if every finite set of its vertices pairwise connected by edges spans a simplex of X. A cycle in a simplicial complex is a subcomplex homeomorphic to the circle S_{1}. The length |γ| of a cycle γ is the number of its edges. We say that a cycle in X has a diagonal if some two nonconsecutive vertices in this cycle are connected by an edge in X.

Let k ≥ 5 denote some natural number. A simplicial complex X is k-large if it is flag and every cycle in X of length 4 ≤ |γ| < k has a diagonal. A simplicial complex is locally k-large if its links at all (nonempty) simplices are k-large. A simplicial complex is k-systolic if it is locally k-large, connected and simply connected. Since the case of k = 6 is particularly important, we abbreviate the term “6-systolic” to an easier to pronounce term systolic.

We view k-largeness as local “curvature” condition for links of a complex, and local k-largeness as a kind of curvature bound from above. We sometimes call local 6-largeness simplicial nonpositive curvature since it yields similar consequences as metric nonpositive curvature. A systolic complex is then the simplicial analogue of what is called a CAT (0) or Hadamard space.

A straightforward consequence of the above definitions is the following.

Lemma 2.1. Let X denote some simplicial complex.

(1) If k ≥ m and X is k-large then X is also m-large.

(2) If k ≥ m and X is k-systolic then X is also m-systolic.

An example of a local-to-global phenomenon related to simplicial nonpositive curvature is the following.

Proposition 2.2 (see Proposition 1 in [JS2]). For k ≥ 6 every k-systolic complex is k-large.

The next result exhibits analogy between systolicity and nonpositive (or even negative) curvature.

Theorem 2.3. (1) (see Theorem 14.1 in [JS2]) If X is a finite dimensional sys-

tolic simplicial complex then X is contractible.

(2) (see Theorem 13 in [JS2]) If X is a 7-systolic simplicial complex then the 1-skeleton of X, equipped with the geodesic metric for which every edge has length 1, is δ-hyperbolic with δ = \frac{5}{2}.

It is interesting to mention that there is no obvious relationship between local 6largeness (simplicial nonpositive curvature) and nonpositive curvature in metric sense. However, k-systolicity for k sufficiently large implies metric nonpositive curvature (see Section 15 in [JS2] for details).

A group is k-systolic if it acts by simplicial automorphisms, properly discontinuously and cocompactly, on a k-systolic simplicial complex. It is systolic, if it is k-systolic with k = 6. As a consequence of Theorem 2.3(2) above we get the following.

Corollary 2.4. Every 7-systolic group is word-hyperbolic.

As it is shown in [JS2] and [Hag2], there exist many k-systolic groups, for every k ≥ 6, in arbitrary dimension. For example, they arise as fundamental groups of certain simplices of groups, which is explained also in the next section (Proposition 3.6 and Proposition 3.7).

2.2. Convexity and saliency. We now turn to describing local convexity phenomena in systolic complexes. We follow Sections 4 and 8 of [JS2]. For a simplicial complex X we denote its link at a face σ by X_{σ}. The next definition of 3-convexity is an equivalent reformulation of the definition given in Section 4 of [JS2].

Definition 2.5 (convex subcomplex). A subcomplex Q in a 6-large simplicial complex X is 3-convex if it is a full subcomplex and if, given any path (v_{1}, v_{2}, v_{3}) in the 1skeleton of X, with v_{1}, v_{3} contained in Q and not connected with an edge in X, we get v_{2} ∈ Q. A subcomplex Y in a systolic complex X is convex if it is connected and for every vertex v ∈ Q the link Q_{v} is 3-convex in the corresponding link X_{v}.

In Section 4 we will prove that convexity is equivalent to geodesic convexity, as defined in the introduction (see Proposition 4.9).

Among various properties of the above combinatorial notion of convexity one of an interest for us is the following (see Lemmas 11.1, 11.2 and Fact 1.3 in [JS2]).

Lemma 2.6. Any convex subcomplex in a systolic complex is full and contractible. In particular, for any k ≥ 6, any convex subcomplex in a k-systolic complex is itself k-systolic.

It turns out that another notion, slightly stronger than convexity, is very usefull in studying systolic complexes and groups. In [JS2] this notion is called strong convexity, but here we will use the term salient, as in [Hag1]. Given a subcomplex Y in a simplicial complex X, its neighbourhood in X, denoted B(Y, X), is the union of all simplices of X that intersect Y.

Definition 2.7 (salient subcomplex). A subcomplex Y in a systolic simplicial complex X is salient if it is connected and for every simplex σ in Y either we have equality Y_{σ} = X_{σ} or there is a simplex τ in X_{σ} such that Y_{σ} = B(τ, X_{σ}).

The next two results indicate important relationships between the notions of salient and convex subcomplexes.

Lemma 2.8 (see Corollary 8 in [JS2]). Every salient subcomplex of a systolic complex X is convex in X.

Theorem 2.9 (see Corollary 9 in [JS2]). Let Y be a convex subcomplex in a systolic complex X. Then its neighbourhood B(Y, X) is a salient subcomplex of X.

We will use the following simplified definition of saliency:

Lemma 2.10. A subcomplex Y in a systolic simplicial complex X is salient iff for every vertex v ∈ Y either Y_{v} = X_{v} or Y_{v} = B(τ, X_{v}) for some simplex τ ⊂ X_{v}.

Proof. We need to show that, if links of Y at vertices are as above then links at other simplices also have the desired form. Let σ be a simplex of Y which is not a vertex, and let σ = v∗ρ. We then have Y_{σ} = (Y_{v})_{ρ}. If Y_{v} = X_{v}, we have also Y_{σ} = X_{σ}, so that Y_{σ} is as desired. If Y_{v} = B(τ, X_{v}), we get Y_{σ} = [B(τ, X_{v})]_{ρ}. Since the neighbourhood of τ in X_{v} has the same links as the corresponding neighbourhood in the universal cover of X_{v}, and the latter neighbourhood is salient by Theorem 2.9, we conclude that the link Y_{σ} is as desired. □

2.3. Balls and spheres. We now turn to describing balls and spheres in systolic complexes. Recall that for any integer n ≥ 1 and any subcomplex Y of a simplicial complex X we define inductively the neighbourhoods B_{n}(Y, X) by B_{1}(Y, X) = B(Y, X) and B_{n+1}(Y, X) = B(B_{n}(Y, X), X).

We state without proof the following easy observations concerning neighbourhoods in simplicial complexes.

Lemma 2.11. Let X be a simplicial complex and Y a subcomplex. Denote by Y^{0} the set of vertices of Y.

(1) For any n ≥ 1 we have B_{n}(Y, X) = ⋃{B_{n}(y, X) : y ∈ Y^{0}}.

(2) Let x be a vertex of X with d(x, Y) = n. Denote by Proj_{Y} (x) the set of those vertices of Y which remain at distance n from x. Then

[B_{n}(Y, X)]_{x} = ∪_{y∈Proj_{Y} (x)}[B_{n}(y, X)]_{x}.

Given a vertex v in a simplicial complex X, the balls in X centered at v are the neighbourhoods B_{n}({v}, X) (we rather denote them by B_{n}(v, X) and we set B_{0}(v, X) = {v}). We also define the distance between two vertices of X as the minimal length of a combinatorial path in the 1-skeleton of X joining the two vertices.

Balls in systolic complexes satisfy many interesting properties, which we start to list in the lemma below. Part (1) of this lemma follows from Lemma 2.8 and Theorem 2.9, while part (2) from Lemma 2.6.

Lemma 2.12. Let X be a systolic simplicial complex and let B = B_{n}(v, X) be any ball in X, for some n ≥ 1. Then

(1) B is a salient (and hence also convex) subcomplex in X.

(2) B is a full subcomplex in X and it is spanned by the set of vertices remaining at distance ≤ n from v.

Let X denote a simplicial complex and let v be a vertex of X. The sphere S_{n}(v, X) is the full subcomplex of X spanned by the set of vertices remaining at distance n from v. Since balls in systolic complexes are full, we get the following.

Corollary 2.13. Let X be a systolic simplicial complex and v a vertex of X. Then for any natural n we have S_{n}(v, X) ⊂ B_{n}(v, X).

We now come to an essential property of balls in systolic complexes. Given a simplex σ in a simplicial complex X, the residue of σ in X (called also the star of σ in X), denoted Res(σ, X), is the subcomplex of X equal to the union of all simplices of X that contain σ.

Lemma 2.14 (see Lemma 14 in [JS2]). Let X be a systolic simplicial complex and v a vertex of X. Then for any natural number n ≥ 1 and any simplex ρ ⊂ S_{n+1}(v, X) the intersection B_{n}(v, X) ∩ Res(ρ, X) is nonempty and it is a single simplex of X.

We will call the simplex B_{n}(v, X) ∩ Res(ρ, X) as above the projection of ρ ⊂ S_{n+1}(v, X) onto S_{n}(v, X). Projections are useful to describe links of balls, as in the following.

Lemma 2.15 (see Corollary 7 in [JS2]). Let x, y be two vertices at distance n in a systolic simplicial complex X, and let σ be the projection of x on the sphere S_{n−1}(y, X). Then [B_{n}(y, X)]_{x} = B(σ, X_{x}).

3. Extra-tilability and local extra-tilability for group actions.

For any group Γ acting on a simplicial complex X and for any simplex σ of X we denote by Γ_{σ} the stabilizer of σ in Γ.

Definition 3.1 (strict fundamental domain). Let Γ denote a group acting by automorphisms on a simplicial complex X. We say that a subcomplex Y ⊂ X is a strict fundamental domain for the action of Γ on X provided the orbit of any simplex of X contains one and only one simplex of Y, and furthermore we have Γ_{σ} = {1} for any simplex σ interior to Y.

Note that the second condition is usually not needed (see Definition 12.7 in [BH]). But the definition above is more appropriate for our purposes and it simplifies the statements.

The following is clear:

Lemma 3.2. Let Γ denote a group acting on a simplicial complex X. Assume that the subcomplex Y ⊂ X is a strict fundamental domain for the action of Γ on X. Then for any simplex σ of Y the subcomplex Y_{σ} ⊂ X_{σ} is a strict fundamental domain for the action of Γ_{σ} on X_{σ}.

Definition 3.3 (chamber-transitive and extra-tilable actions). Let X be a connected simplicial complex, and let Γ be a group acting on X. For any subcomplex Y ⊂ X, we denote by Γ(Y) < Γ the subgroup generated by the stabilizers Γ_{σ}, with σ a maximal simplex of ∂Y. We say that Y is a tile for (Γ, X) if Y is a strict fundamental domain for Γ(Y).

We say that the action of Γ on X is chamber transitive if it admits a strict fundamental domain consisting in a single simplex Δ. We say that the action is extra-tilable if for any simplex σ of X the neighbourhood B(σ, X) is a tile.

We say that the action of Γ on X is locally extra-tilable if for any simplex σ the action of Γ_{σ} on X_{σ} is extra-tilable.

A result analogous to the following is established in Proposition 18 from [JS2]. For the convenience of the reader we recall the argument.

Theorem 3.4. Let X denote some systolic simplicial complex. Let Γ be a group acting on X. Assume that the action is locally extra-tilable. Then every salient subcomplex is a tile.

To prove this theorem we use the theory of simple complexes of groups, as developed in [BH], Chapter II.12.

Proof. Let Y ⊂ X denote any salient subcomplex. We first define a simple complex of groups G^{Y} over Y with local groups Γ_{σ}^{Y}. For any maximal simplex σ of ∂Y we set Γ_{σ}^{Y} = Γ_{σ}. For any other simplex σ of ∂Y we let Γ_{σ}^{Y} denote the subgroup of Γ_{σ} generated by the Γ_{τ} ’s, for all maximal simplices τ of ∂Y containing σ. For any simplex σ in the interior of Y we set Γ_{σ}^{Y} = {1}. Clearly when σ_{2} ⊂ σ_{1} ⊂ Y we have Γ_{σ_{1}}^{Y} ⊂ Γ_{σ_{2}}^{Y}.

Consider a natural (simple) morphism m^{Y} : G^{Y} → Γ(Y) given by the canonical inclusions Γ_{σ}^{Y} ⊂ Γ(Y), and note that this morphism is injective on local groups (see the end of Definition 12.11 in [BH]). Thus the complex of groups G^{Y} has a development corresponding to the morphism m^{Y}. This means that there exists an action of Γ(Y) on a (multi)simplicial complex \overline{X} containing Y as a strict fundamental domain, such that the stabilizer of any simplex σ ⊂ Y ⊂ \overline{X} is precisely Γ_{σ}^{Y} (see Theorem 12.18 in [BH]). Since by definition the groups Γ_{σ}^{Y} generate Γ(Y), the space \overline{X} is connected ([BH], Proposition 12.20 (2)). To avoid confusions we denote Γ(Y) by \overline{Γ(Y)} when it acts on \overline{X}.

Since Γ_{σ}^{Y} ⊂ Γ_{σ}, there is a natural map sending a point \overline{γ}y ∈ \overline{X} to γy ∈ X, where \overline{γ} = γ ∈ \overline{Γ(Y)} = Γ(Y) and y ∈ Y. In order to get conclusion it is sufficient to show that this map \overline{X} → X is an isomorphism. In fact, since X is simply-connected, it is enough to show that \overline{X} → X is a covering. By homogeneity, this reduces to checking that \overline{X} → X is a local isomorphism at simplices σ of ∂Y.

Let σ be any simplex of ∂Y. By Lemma 3.2, the action of the group \overline{Γ(Y)}_{σ} = Γ_{σ}^{Y} on the link \overline{X}_{σ} admits Y_{σ} as a strict fundamental domain. Furthermore, for any simplex τ of Y_{σ} we see that the stabilizer of τ in \overline{Γ(Y)}_{σ} is Γ_{σ∗τ}^{Y}, where σ ∗ τ is the join of σ with τ, i.e. the simplex spanned by the union of σ and τ. On the other hand, since Y is salient in X, there exists a simplex ρ in Y such that Y_{σ} = B(ρ, X_{σ}). By assumption the action of Γ is locally extra-tilable, so B(ρ, X_{σ}) is a tile for the action of Γ_{σ} on X_{σ}. This means that the subgroup Γ_{σ}(B(ρ, X_{σ})) < Γ_{σ} admits B(ρ, X_{σ}) = Y_{σ} as a strict fundamental domain. By definition Γ_{σ}(B(ρ, X_{σ})) is generated by all stabilizers (Γ_{σ})_{τ}, for all maximal simplices τ of the boundary of Y_{σ}. But (Γ_{σ})_{τ} = Γ_{σ∗τ}, and the join σ ∗ τ varies all over the set of maximal simplices of ∂Y containing σ. Thus in fact Γ_{σ}(B(ρ, X_{σ})) = Γ_{σ}^{Y}.

We observe that the same group Γ_{σ}^{Y} acts on two simplicial complexes (\overline{X}_{σ} and X_{σ}) with the same strict fundamental domain (namely Y_{σ}). Thus the natural map \overline{X}_{σ} → X_{σ} is an equivariant isomorphism, provided we prove that stabilizers of simplices in the fundamental domain are the same for the two actions (see for example Proposition 12.20 (1) in [BH]). Let τ denote some simplex of Y_{σ}. We have noticed that the stabilizer of τ in \overline{Γ(Y)}_{σ} is Γ_{σ∗τ}^{Y}. Moreover, we clearly have Γ_{σ∗τ}^{Y} ⊂ Γ_{σ∗τ} ∩Γ_{σ}^{Y} = (Γ_{σ}^{Y})_{τ}. It remains to check the reverse inclusion. There is nothing to prove if τ is in the interior of Y_{σ}, for in this case (Γ_{σ}^{Y})_{τ} = {1}. Assume that τ ⊂ ∂Y_{σ}. By saliency there exists a simplex ρ′ in the interior of Y such that Y_{σ∗τ} = B(ρ′, X_{σ∗τ}). As we have seen, Y_{σ∗τ} is a strict fundamental domain for the action of Γ_{σ∗τ}^{Y} on X_{σ∗τ}. Thus for any γ ∈ Γ_{σ}^{Y} such that γτ = τ, there exists a (unique) γ′ ∈ Γ_{σ∗τ}^{Y} such that γ^{′−1}γρ′ ⊂ Y_{σ∗τ}. Since γ^{′−1}γ ∈ Γ_{σ}^{Y} and Y_{σ} is a strict fundamental domain for the action of Γ_{σ}^{Y}, we have γ^{′−1}γρ′ = ρ′ and thus in fact γ = γ′, since ρ′ is interior to Y_{σ} too. □

An effective source of groups acting in an extra-tilable (and locally extra-tilable) way is a construction of appropriate simplices of groups given in [JS2] and [Hag2]. We recall briefly the outcome of this construction. For the general theory of simple complexes of groups we refer the reader to [BH], Chapter II.12. In particular, the notion of a local development is explained there in Construction 12.24, p. 387.

Definition 3.5. Let G be a simplex of groups, let Δ denote the underlying simplex, and for any face σ of Δ let G_{σ} denote the local group of G at σ. A simplex of groups is locally k-large if all of its local developments are k-large simplicial complexes. A simplex of groups is locally extra-tilable if the action of every local group G_{σ} on the local development at σ is extra-tilable. When the local group G_{Δ} is trivial, we say that the simplex of groups is ∂-supported.

Proposition 3.6. Given an integer k ≥ 6, let G be a ∂-supported, locally k-large simplex of finite groups, and let Γ denote its fundamental group. Then G is developable, its universal cover is k-systolic and the action of Γ on the universal cover is chamber-transitive and properly discontinuous. In particular, if k ≥ 7 then Γ is word-hyperbolic. If furthermore G is locally extra-tilable, then the action of Γ on the universal cover is locally extra-tilable.

Proof. The developability of G follows by Theorem 15 in [JS2]. The fact that any maximal simplex Δ in the universal cover is then a strict fundamental domain for Γ is clear (Δ has trivial stabilizer since G is ∂-supported). Proper discontinuity follows from finiteness of local groups. Local k-argeness and local extra-tilability follow from the corresponding assumptions for local developments. Consequently, under the local 7-largeness assumption, the universal cover is 7-systolic and Γ is then word-hyperbolic by Corollary 2.4. □

Locally 7-large and locally extra-tilable simplices of finite groups are the main objectives of our study in this paper. The next proposition shows the existence of great many of such simplices of groups.

Proposition 3.7 (Propostion 21 in [JS2]). Let Δ be a simplex (of arbitrary dimension) and suppose that for any codimension 1 face s of Δ we are given a finite group A_{s}. Then for any k ≥ 6 there exists a ∂-supported simplex of finite groups G over Δ, with local groups G_{σ}, which is locally k-large, locally extra-tilable and such that G_{s} = A_{s} for each codimension 1 face s in Δ.

4. Systolic complexes: further results.

In this section we derive useful properties of geodesics in systolic complexes.

Definition 4.1. A combinatorial path of X is a sequence γ = (v_{0}, …, v_{n}) of vertices of X such that for each i = 0, …, n − 1 the vertices v_{i}, v_{i+1} span an edge of X. The path γ is said to have length n. The endpoints of γ are v_{0} and v_{n}. Given combinatorial paths γ = (v_{0}, …, v_{n}) and γ′ = (v_{n}, …, v_{n+m}), we denote by γ.γ′ their concatenation (v_{0}, …, v_{n}, v_{n+1}, …, v_{n+m}).

The combinatorial distance between two vertices v, w is the minimum of the lengths of paths with endpoints v, w; we will denote it by d(v, w). A combinatorial path with endpoints v, w minimizing the length is called a geodesic of X between v and w.

4.1. Ladders and bigons.

Definition 4.2 (ladder). A ladder is an ordered pair (γ = (v_{0}, …, v_{n}), γ′ = (v′_{0}, …, v′_{n})) of (combinatorial) geodesics of X (with the same length), such that:

(1) for each i = 0, …, n − 1 the subset {v_{i}, v′_{i}, v_{i+1}} spans a simplex,

(2) for each i = 1, …, n the subset {v′_{i−1}, v′_{i}, v_{i}} spans a simplex. Note that when (γ, γ′) is a ladder, in general (γ′, γ) is not a ladder.

Definition 4.3 (bigon). A bigon is an ordered pair (γ = (v_{0}, …, v_{n}), γ′ = (v′_{0}, …, v′_{n})) of (combinatorial) geodesics of X (with the same length) such that v_{0} = v′_{0} and d(v_{n}, v′_{n}) ≤ 1. We say that a bigon is k-thin whenever we have d(v_{i}, v′_{i}) ≤ k for i = 0, …, n.

Clearly, a ladder with v_{0} = v′_{0} is a 1-thin bigon.

Lemma 4.4. Let X denote some systolic complex. Let γ = (v_{0}, …, v_{n}) denote some geodesic of X, and let w denote some vertex of X such that d(v_{0}, w) = n and d(w, v_{n}) ≤ 1. Then there exists a geodesic γ′ = (v′_{0}, …, v′_{n}), with v′_{0} = v_{0} and v′_{n} = w, such that (γ, γ′) is a ladder.

Proof. The lemma is obviously true for n = 0. We argue by induction on n ≥ 1. For n = 1 the flagness of X implies the statement. For n > 1, if w = v_{n}, we may take γ′ = γ. If {w, v_{n}} spans an edge e in X, let u denote a vertex of the projection of e ⊂ S_{n}(v_{0}) onto S_{n−1}(v_{0}) (see Lemma 2.14 and the definition after it). Then {u, w, v_{n}} spans a simplex. Since v_{n−1} and u are in the projection of v_{n} onto S_{n−1}(v_{0}), we see that {v_{n−1}, u, v_{n}} spans a simplex too. By induction, there is a geodesic γ′′ from v_{0} to u such that ((v_{0}, v_{1}, …, v_{n−1}), γ′′) is a ladder. It follows that (γ, γ′′.(u, w)) is also a ladder, and the lemma holds for γ′ = γ′′.(u, w). □

Corollary 4.5. Let X denote some systolic complex. Let γ = (v_{0}, …, v_{n}) denote some geodesic in X, with n ≥ 1. Let u denote any vertex of X such that d(v_{0}, u) = n − 1 and d(u, v_{n}) = 1. Then there exists a geodesic γ′ from v_{0} to v_{n}, such that (γ, γ′) is a ladder, and the two last vertices of γ′ are u and v_{n}.

Proof. Since u and v_{n−1} belong to the projection of v_{n} ∈ S_{n}(v_{0}) onto S_{n−1}(v_{0}), we know that {u, v_{n−1}} spans a simplex. By Lemma 4.4, there exists a geodesic γ′_{1} from v_{0} to u such that ((v_{0}, v_{1}, …, v_{n−1}), γ′_{1}) is a ladder. Then (γ, γ′_{1}.(u, v_{n})) is a ladder, and γ′_{1}.(u, v_{n}) has the required properties. □

Under assumption of 7-systolicity we will get stronger properties of geodesics. To do this, we need a preparatory lemma.

Lemma 4.6. Let X denote some 7-large complex. Let σ, τ denote two simplices of X. Assume that τ ∩ B(σ, X) = ∅. Then B(σ, X) ∩ B(τ, X) is either empty or a simplex.

Proof. Note that the hypothesis τ ∩ B(σ, X) = ∅ in fact implies σ ∩ B(τ, X) = ∅. Assume that s, t are two distinct vertices of B(σ, X) ∩ B(τ, X). Then each of them is disjoint from σ and τ, but adjacent to some vertices of σ, and also to some vertices of τ.

Assume that s, t are adjacent to the same vertex p of σ. If they are also adjacent to the same vertex q of τ then (p, s, q, t) is a cycle in X of length 4. Since p and q are not adjacent by the assumption, 7-largeness of X implies that s and t are adjacent. If s, t are adjacent to distinct vertices q, r of τ then (p, s, q, r, t) is a cycle in X of length 5, and (s, t) is the only possible diagonal for this cycle. Again, by 7-largeness of X, s and t are adjacent.

Now consider the case when s, t are adjacent to distinct vertices in both σ and τ. An argument similar as above shows that s and t are adjacent in X. The lemma follows then by flagness of X. □

Proposition 4.7 (see Lemma 3.4 in [Hag1]). In a 7-systolic complex all bigons are 1-thin.

Proof. Let (γ, γ′) denote some bigon in X. Let v denote the common origin of γ, γ′, and let w, w′ denote their endpoints. We prove that (γ, γ′) is 1-thin by induction on n = d(v, w) = d(v, w′).

There is nothing to prove if n ≤ 1, so let us assume that n ≥ 2. Let then s, s′ denote the vertices on γ, γ′ just before w, w′. By induction it suffices to prove that d(s, s′) ≤ 1.

By Lemma 4.4 there is a geodesic γ′′ from v to w′ such that (γ, γ′′) is a ladder. Let t, p denote the vertices of γ′′ at distance n−2 and n−1 from v. By Corollary 4.5 there is a geodesic γ′_{1} from v to s′ such that (γ′′, γ′_{1}.(s′, w′)) is a ladder. Let t′ denote the vertex of γ′_{1} just before s′. Denote by σ the simplex spanned by {w, w′} (of dimension 1 or 0), and by τ the simplex spanned by {t, t′} (τ is well-defined because (γ′′, γ′_{1}.(s′, w′)) is a ladder). Since t, t′ ∈ S_{n−2}(v_{0}) and w, w′ ∈ S_{n}(v_{0}), we have τ ∩ B(σ, X) = ∅.

By Lemma 4.6 the intersection B(σ, X) ∩ B(τ, X) is a simplex. Since this intersection contains s and s′, t follows that d(s, s′) ≤ 1. This completes the proof. □

Note that the previous Proposition implies, together with a criterion of Papasoglu (see [P]), that 7-systolic groups are hyperbolic (see [Hag1]).

4.2. Geodesic convexity. In this subsection we link the (global) notion of geodesic convexity (see Introduction) to the (local) notion of convexity (see Definition 2.5).

Lemma 4.8. Let X denote some systolic complex, and let Y ⊂ X denote any convex subcomplex. Assume that (γ = (v_{0} = v, …, v_{n} = w), γ′ = (v′_{0} = v, …, v′_{n} = w)) is a bigon of X such that all vertices of γ are in Y. Then all vertices of γ′ are in Y.

Proof. We argue by induction on the length n.

Let (γ = (v_{0} = v, …, v_{n}, v_{n+1} = w), γ′ = (v′_{0} = v, …, v′_{n}, v′_{n+1} = w)) denote a bigon of X such that all vertices of γ are in Y. By Corollary 4.5 there is a geodesic γ_{1} = (u_{0} = v_{0}, u_{1}, …, u_{n} = v′_{n}) from v_{0} to v′_{n} such that (γ, γ_{1}.(v′_{n}, v_{n+1})) is a ladder. We claim that all vertices of γ_{1} are in Y. To see this, we look first at u_{1}. If u_{1} = v_{1} then u_{1} ∈ Y. Otherwise (v_{0}, u_{1}, v_{2}) is a path of X_{v_{1}} with endpoints in Y_{v_{1}}. Since v_{0}, v_{2} are at distance 2 in X, and hence also in X_{v_{1}}, by 3-convexity of Y_{v_{1}} in X_{v_{1}} we have u_{1} ∈ Y. Furthermore, we get two length n geodesics (u_{1}, v_{2}, …, v_{n}, v_{n+1}) and (u_{1}, …, u_{n}, v_{n+1}), such that all vertices of the first geodesic belong to Y. By induction, all vertices u_{1}, …, u_{n} belong to Y and thus all vertices of γ_{1} are in Y.

Now, we get another pair of length n geodesics γ_{1} and (v′_{0}, …, v′_{n}), with common endpoints, such that all vertices of the first geodesic belong to Y. By induction, all vertices v′_{1}, …, v′_{n} belong to Y, which finishes the proof.

□

Proposition 4.9. Let X denote some systolic complex, and let Y ⊂ X denote a (non empty) subcomplex. Then Y is convex iff it is geodesically convex.

Proof. First assume that Y is a geodesically convex subcomplex. Then Y is clearly connected. Let us check now that Y is locally 3-convex. Let v denote some vertex of Y, and let (v_{1}, v_{2}, v_{3}) denote any path of the link X_{v}, such that v_{1}, v_{3} ∈ Y_{v} and v_{1}, v_{3} are not connected by an edge in X_{v}. Then v_{1}, v_{3} are distinct vertices of X, and their combinatorial distance cannot be 1 (for otherwise there would be an edge in X joining v_{1}, v_{3} and, by flagness of X, this edge would belong to X_{v}). Thus (v_{1}, v_{2}, v_{3}) is a geodesic of X. By geodesic convexity we then have v_{2} ∈ Y, and since Y is full by assumption, we also get v_{2} ∈ Y_{v}. Consequently, Y is convex.

To prove the converse, assume that Y is a convex subcomplex of X, i.e. it is connected and locally 3-convex. By Lemma 2.6 Y is then full. By Lemma 4.8 we need to prove that for any two vertices p, q in Y there exists a geodesic of X, from p to q, all of whose vertices belong to Y.

Let p be some vertex in Y. Put Y_{0} = {p}, and for each integer k ≥ 1 we define inductively Y_{k} as the union of all simplices of Y that intersect Y_{k−1}. Since Y is connected, it is the union of the subcomplexes Y_{k}. Thus it is enough to prove that for each integer k ≥ 0 and each vertex q ∈ Y_{k}, there exists a geodesic of X, from p to q, all of whose vertices are contained in Y. This fact is straightforward for k = 0 or k = 1, and we argue by induction on k ≥ 1.

Let q denote some vertex of Y_{k+1}. There exists a vertex q′ ∈ Y_{k} such that {q, q′} is a simplex of Y. In particular |d(p, q)−d(p, q′)| ≤ 1, where d denotes the combinatorial distance in X. By induction, there is a geodesic γ′ from p to q′ all of whose vertices are contained in Y.

If d(p, q) > d(p, q′), then the concatenation γ′.(q′, q) is a geodesic from p to q all of whose vertices are contained in Y. If d(p, q′) > d(p, q), then q is on some geodesic γ from p to q′. Since γ and γ′ have the same endpoints, by Lemma 4.8 all the vertices of γ are in Y. In particular, the initial part of γ between p and q is a geodesic with all vertices in Y.

Assume now that d(p, q) = d(p, q′). By Lemma 4.4, there exists a geodesic γ from p to q such that (γ′, γ) is a ladder. Let q′′ denote the vertex of γ just before q, and let γ′′ be the initial part of γ between p and q′′. Then q′′ is linked to q′ and γ′′.(q′′, q′) is a geodesic from p to q′. It follows by Lemma 4.8 that all the vertices of γ′′, and thus also of γ, are in Y, which completes the proof. □

Corollary 4.10. The balls of a systolic complex are geodesically convex.

Proof. By Lemma 2.12 the balls are salient and hence convex. Thus, by Proposition 4.9, they are geodesically convex. □

5. Quasi-convexity in 7-systolic complexes

In this section we show that each quasi-convex subcomplex in a 7-systolic complex is at finite Hausdorff distance of a convex subcomplex (Theorem 5.5).

Lemma 5.1 (Y-lemma). Let v denote a vertex in a systolic complex X. Let v_{1}, v_{2} denote vertices at distance n from v and with d(v_{1}, v_{2}) = d ≤ n. Then there is a geodesic of length n − d with origin v that extends to a geodesic to either of the vertices v_{i}.

Proof. We argue by induction on d. The result is obvious for d = 0.

Let (x_{0} = v_{1}, x_{1}, …, x_{d} = v_{2}) denote a geodesic. By Corollary 4.10 this geodesic is contained in B_{n}(v, X). If d(v, x_{1}) = n we choose a vertex w_{1} such that d(v, w_{1}) = n−1 and w_{1}, x_{0}, x_{1} span a simplex. If d(v, x_{1}) = n − 1 we set w_{1} = x_{1}. We define similarly w_{2} (where now v_{2} plays the role of v_{1}).

We claim that d(w_{1}, w_{2}) ≤ d − 1.

To prove the claim we consider two cases. In the first case we assume that the geodesic (x_{0}, x_{1}, …, x_{d}) does not enter the ball B_{n−1}(v, X). In this case there is a sequence (y_{1} = w_{1}, y_{2}, …, y_{d} = w_{2}) of vertices of the sphere S_{n−1}(v, X) such that y_{k}, x_{k−1}, x_{k} span a simplex. Then the vertices y_{k}, y_{k+1} span a simplex, since they both belong to the projection of x_{k} on the sphere S_{n−1}(v, X). Consequently, we have d(w_{1}, w_{2}) ≤ d − 1.

In the second case we assume that the geodesic (x_{0}, x_{1}, …, x_{d}) enters the ball B_{n−1}(v, X), so there are integers 1 ≤ i ≤ j ≤ d − 1 such that

d(v, x_{0}) = d(v, x_{1}) = ⋯ = d(v, x_{i−1}) = n = d(v, x_{j+1}) = ⋯ = d(v, x_{d})

and

d(v, x_{i}) = d(v, x_{j}) = n − 1.

Observe that there is a sequence (y_{1} = w_{1}, y_{2}, …, y_{i} = x_{i}) of vertices of the sphere S_{n−1}(v, X) such that y_{k}, x_{k−1}, x_{k} span a simplex. Similarly, there is a sequence (y_{j} = x_{j}, y_{j+1}, …, y_{d−1} = w_{2}) of vertices of the sphere S_{n−1}(v, X) such that y_{k}, x_{k+1}, x_{k} span a simplex. As in the previous case, any two subsequent vertices y_{K}, y_{k+1} in any of the sequences (y_{1}, y_{2}, …, y_{i}) and (y_{j}, y_{j+1}, …, y_{d−1}) span a simplex, hence we have d(w_{1}, w_{2}) ≤ (i − 1) + (j − i) + (d − j − 1) = d − 2. Thus the claim follows.

By induction there is a geodesic of length (n − 1) − d(w_{1}, w_{2}) ≥ n − d with origin v that extends to a geodesic to either of the vertices w_{i}. Both of those geodesics can be extended to vertices v_{i}, which finishes the proof. □

The next two results now require the assumption of 7-systolicity.

Lemma 5.2. Let X denote a 7-systolic complex. Let (v_{0}, v_{1}, ⋯, v_{n}), (v_{0}, v′_{1}, ⋯, v′_{n}) denote two geodesics with the same origin and such that d(v_{1}, v′_{1}) = 2. Then n < 2 + d(v_{n}, v′_{n}).

Proof. We argue by contradiction. Suppose that d(v_{n}, v′_{n}) ≤ n − 2 (in particular n ≥ 2). By Lemma 5.1, there is a geodesic (v_{0}, x, y) which extends to geodesics from v_{0} to either of the vertices v_{n} and v′_{n}. Since in 7-systolic complexes geodesic bigons are 1-thin (see Proposition 4.7), we have d(x, v_{1}) ≤ 1, d(x, v′_{1}) ≤ 1, d(y, v_{2}) ≤ 1 and d(y, v′_{2}) ≤ 1.

By Proposition 2.2 the complex X is 7-large: we now use this property to find a contradiction. First, note that there is no edge between v′_{1} and v_{2}. Indeed, if there is such an edge then (v_{0}, v_{1}, v_{2}, v′_{1}, v_{0}) is a cycle of length 4 in X without diagonals. Similarly, there is no edge between v_{1} and v′_{2}. This also means that v_{2} ≠ v′_{2}. Now, note that there is no edge between v_{2} and v′_{2}. Indeed, if there is such an edge then (v_{0}, v_{1}, v_{2}, v′_{2}, v′_{1}, v_{0}) is a cycle of length 5 in X without diagonals. In particular, we have y ≠ v_{2} and y ≠ v′_{2}. But then (v_{0}, v_{1}, v_{2}, y, v′_{2}, v′_{1}, v_{0}) is a cycle of length 6 in X without diagonals, contradiction. □

Corollary 5.3. Let v_{1}, v_{2}, x be vertices of a 7-systolic complex X, and suppose that d(v_{1}, v_{2}) ≤ d and d(v_{1}, x) = d(v_{2}, x) = n ≥ d + 2. Denote by σ_{1}, σ_{2} the projections of the vertex x on the spheres S_{n−1}(v_{1}, X) and S_{n−1}(v_{2}, X) respectively. Then σ_{1} ∪ σ_{2} spans a simplex of X.

Proof. Since X is flag, it is enough to check that, given any vertices w_{1} ∈ σ_{1} and w_{2} ∈ σ_{2}, these vertices either coincide or are connected with an edge of X. We argue by contradiction. Consider vertices w_{1} ∈ σ_{1}, w_{2} ∈ σ_{2} such that d(w_{1}, w_{2}) = 2. Then Lemma 5.2 applies to geodesics of length n from v to v_{1}, v_{2} passing through w_{1}, w_{2} respectively. We deduce that d(v_{1}, v_{2}) > n − 2 ≥ d, which contradicts the assumption. □

Definition 5.4 (quasi-convex subcomplex). Given K > 0, we say that a subcomplex Y in a connected simplicial complex X is K-quasi-convex whenever the following holds: for any geodesic (x_{0}, …, x_{n}) in X such that x_{0} ∈ Y, x_{n} ∈ Y we have d(x_{i}, Y) ≤ K for 0 ≤ i ≤ n. A subcomplex Y is quasi-convex in X if it is K-quasi-convex for some K.

Theorem 5.5. Let X denote a locally finite dimensional 7-systolic complex. Let Y ⊂ X be a K-quasi-convex subcomplex, for some K > 0. Then there exists an integer n(K) such that for every n ≥ n(K) the neighbourhood B_{n}(Y, X) is salient.

Proof. Since X is 7-systolic, it is word-hyperbolic (see Theorem 2.3(2)). Thus X admits quasi-projections on quasi-convex subcomplexes (see e.g. Proposition 3.11 on p. 463 in [BH]). In particular, there exists an integer d = d(K) ≥ 0 such that for any vertex x in X and any vertices v_{1}, v_{2} in Y, if d(x, v_{1}) = d(x, Y) = d(x, v_{2}) then we have d(v_{1}, v_{2}) ≤ d.

Now, we set n(K) = max (K, d(K) + 2). Since n(K) ≥ K and Y is K-quasi-convex, it follows that for any n ≥ n(K) the neighbourhood B_{n}(Y, X) is connected. To prove that B_{n}(Y, X) is salient, it remains to check its links at vertices (see Lemma 2.10).

Let x be a vertex of B_{n}(Y, X). If d(x, Y) < n, we have [B_{n}(Y, X)]_{x} = X_{x}. Thus, it remains to consider the case when d(x, Y) = n. In this case, denote by Proj_{Y} (x) the set of vertices of Y at distance n from x. By the properties of quasi-projections, the set Proj_{Y} (x) has diameter at most d(K). For each y ∈ Proj_{Y} (x), denote by σ_{y} the projection of x on the sphere S_{n−1}(y, X). We claim that the set of simplices {σ_{y} : y ∈ Proj_{Y} (x)} spans a simplex of X. Indeed, since n ≥ n(K) ≥ d(K) + 2, it follows from Corollary 5.3 that any pair of simplices from this set spans a simplex. The claim then follows by flagness of X, and by the fact that X is locally finite dimensional (the latter implies that the set {σ_{y} : y ∈ Proj_{Y} (x)} is finite, even if Proj_{Y} (x) is infinite). We denote by τ the simplex of X spanned by the set {σ_{y} : y ∈ Proj_{Y} (x)}.

Observe that, again due to flagness of X, the simplex τ belongs to the link X_{x}. Moreover, using Lemma 2.11 and Lemma 2.15 we get

[B_{n}(Y, X)]_{x} = ⋃_{y∈Proj_{Y} (x)}[B_{n}(y, X)]_{x} = ⋃_{y∈Proj_{Y} (x)} B(σ_{y}, X_{x}) = B(τ, X_{x}).

Thus B_{n}(Y, X) is salient. □

Definition 5.6. Let Γ be a word-hyperbolic group. A subgroup H < Γ is quasiconvex if H is a quasiconvex subset in the Cayley graph C(Γ, S) for some finite generating set S.

Recall that, since Γ is word-hyperbolic, quasi-convexity does not depend on the choice of the generating set S ([BH], Corollary 3.6, p. 461).

Remark 5.7. When H is a quasi-convex subgroup of a word-hyperbolic group Γ, and Γ acts discretely cocompactly on a simplicial complex X, then for any compact subcomplex D ⊂ X the union Y = ∪_{h∈H}hD is a quasiconvex subcomplex of X.

Indeed, for any vertex v_{0} ∈ D, the map γ ↦ γv_{0} is a quasi-isometry of Γ onto X (admitting a quasi-inverse). It follows that the orbit H{v_{0}} is quasiconvex, because quasi-geodesics of Γ are at finite Hausdorff distance of a geodesic ([BH], Theorem 1.7, p. 401). Now the quasi-convexity of Y follows, since by compactness of D and hyperbolicity of X any geodesic with endpoints in Y is at finite Hausdorff distance of a geodesic with endpoints in some orbit H{v_{0}} (v_{0} a vertex of D).

Observe that in the next corollary Γ is a 7-systolic group, and hence it is wordhyperbolic (see Corollary 2.4).

Corollary 5.8. Let X be a locally compact 7-systolic complex and let Γ be a group acting discretely and cocompactly on X. Then any quasiconvex subgroup H is convex cocompact in the following sense: there exists a convex (even salient) subcomplex Y such that H preserves Y and H acts on Y discretely and cocompactly.

Proof. Since H is quasiconvex for any fixed vertex v_{0} the subcomplex Z = H.{v_{0}} is quasiconvex. Thus by Theorem 5.5 for some large n the subcomplex Y = B_{n}(Z, X) is salient. Since Z is H-invariant, so is Y. And since H is cocompact on Y and balls in X are compact, H is also cocompact on Y.

□

6. QCS for 7-systolic simplices of groups

In this section we prove the main result of the paper, Theorem 6.1. We also give some explicit examples of groups to which this theorem applies.

Theorem 6.1. Let G be a locally 7-large, locally extra-tilable simplex of finite groups. Then any quasi-convex subgroup of the fundamental group of G is separable.

Proof. Let X denote the universal cover and Γ the fundamental group of G. By Proposition 3.6 the simplicial complex X is then 7-systolic and Γ acts on it properly discontinuously, cocompactly, and in a locally extra-tilable way.

Let H be any quasi-convex subgroup of Γ. We want to show that H is an intersection of finite index subgroups. That is, we need to show that for any element g ∈ Γ \ H there is a finite index subgroup Γ′ < Γ such that H < Γ′ and g ∉ Γ′.

Let Δ be some top-dimensional simplex of X. Consider the subcomplex Z of X being the union of all simplices in the orbit of Δ under H. Z is then a quasi-convex subcomplex of X by Remark 5.7. Let g be any element of Γ \ H. There exists an integer n such that the subcomplex Y = B_{n}(Z, X) contains the simplex gΔ. By Theorem 5.5, choosing n larger if necessary, we may assume that Y is salient. It is also clear that H preserves Y, and that H is cocompact on Y (by local finiteness of X). By Theorem 3.4, Y is a strict fundamental domain for the subgroup Γ(Y) ⊂ Γ generated by the stabilizers of maximal simplices inside ∂Y. Since H preserves Y, it acts by conjugation on this family of stabilizers. Thus H acts by conjugation on Γ(Y). Let Γ′ ⊂ Γ denote the semi-direct product of Γ(Y) and H. We claim that Γ′ is a finite index subgroup of Γ and that Γ′ does not contain g.

First, since Γ acts simply-transitively on the set C of top dimensional simplices in X, the index [Γ : Γ′] is equal to the number of orbits of Γ′ in C. Since Y is a (strict) fundamental domain for Γ(Y), every such Γ′-orbit meets the set C(Y) of top dimensional simplices contained in Y. Since H is cocompact on Y, it is cofinite on C(Y), and thus there are finitely many Γ′-orbits in C, so [Γ : Γ′] < ∞.

We turn to proving that g ∉ Γ′. If, on the contrary, we have g ∈ Γ′, then g = γh for some γ ∈ Γ(Y) and h ∈ H. We also have gΔ = γhΔ, and thus both hΔ and γ(hΔ) are contained in Y. Since Y is a strict fundamental domain for Γ(Y), and since γ ∈ Γ(Y), we have γ(hΔ) = hΔ. This implies that γ = 1, because top-dimensional simplices of X have trivial stabilizers under Γ. Consequently, g = h and thus g ∈ H, which contradicts the assumption. □

As we have already mentioned, simplices of groups satisfying the assumptions of the above theorem exist in abundance at every dimension (Proposition 3.7). As it is shown in [JS2] (Corollary 23.1), the fundamental groups of simplices of groups as above may have arbitrarily large (virtual) cohomological dimension.

In the remaining part of this section we will show explicit examples in dimension 2.

Example 6.2 (some triangular Coxeter groups). Given positive integers m_{1}, m_{2}, m_{3} such that \frac{1}{m_{1}} + \frac{1}{m_{2}} + \frac{1}{m_{3}} < 1 consider some hyperbolic triangle T (m_{1}, m_{2}, m_{3}) with angles \frac{π}{m_{1}} , \frac{π}{m_{2}} , \frac{π}{m_{3}} . Let W (m_{1}, m_{2}, m_{3}) denote the group of isometries of the hyperbolic plane generated by the reflections in the sides of T (m_{1}, m_{2}, m_{3}), and let

X = ⋃_{w∈W (m_{1},m_{2},m_{3})} wT (m_{1}, m_{2}, m_{3})

denote the associated tesselation of ℍ^{2}. It is clear that T (m_{1}, m_{2}, m_{3}) is a strict fundamental domain for the action of W (m_{1}, m_{2}, m_{3}) on X.

The links of vertices in X of corresponding types are 2m_{i}-gons, so X is 7-systolic iff each m_{i} is ≥ 4. Moreover, the stabilizers of vertices are the dihedral groups D_{2m_{i}} of order 2m_{i}, with standard actions on the corresponding links. It is then easy to observe that and the action of W (m_{1}, m_{2}, m_{3}) is locally extra-tilable iff each m_{i} is dividible by 6. By Theorem 6.1, we deduce that the Coxeter group W = ⟨t_{1}, t_{2}, t_{3}|t_{1}^{2}, t_{2}^{2}, t_{3}^{2}, (t_{1}t_{2})^{6k_{3}}, (t_{2}t_{3})^{6k_{1}}, (t_{1}t_{3})^{6k_{2}}⟩ is QCS (for any choice of positive integers k_{1}, k_{2}, k_{3}).

Any surface group is commensurable to a group W as in Example 6.2. Since the property QCS is commensurability invariant (see for example Lemma 7.5 in [HaW]) we have a new proof of the following result (compare [Sc]).

Corollary 6.3. Any surface group is QCS.

Before giving a somewhat thicker two-dimensional example, we describe an extratilable, chamber-transitive action on a graph. We refer to [BH] for the vocabulary on simple complexes of groups.

Example 6.4 ((Z_{2}, Z_{3})-group). Let E be the 1-simplex with vertices v_{1}, v_{2}. Denote by F the simplex of groups over E with vertex groups G_{v_{1}} = Z_{2} and G_{v_{2}} = Z_{3}, and with G_{E} = {1} (so that F is ∂-supported). We will describe a morphism m : F → T to certain group T of order 216.

Let Π be the euclidean plane tesselated by regular hexagons. Denote by a, b, c, d, e, f the vertices (in cyclic order) of a chosen hexagon in Π. Consider the group A = Z ⊕Z of automorphisms of Π generated by translations through vectors 6 · \overrightarrow{ac} and 6 · \overrightarrow{ae}. Then the quotient Π/A is a torus tesselated by 36 hexagons.

For every vertex u of Π rotations through angles 2π/3 and 4π/3 around u are clearly automorphisms of Π. Since the group A is preserved by conjugation through these rotations, they are also automorphisms of the quotient Π/A. Similarly, for a center s of any edge ϵ of Π, reflection with respect to the straight line containig s and orthogonal to ϵ is an automorphism of Π that induces an automorphism of the quotient Π/A. Let T be the group of automorphisms of Π/A generated by rotations and reflections as above.

Let P be the barycentric subdivision of the 1-skeleton of Π/A. Then T acts by automorphisms on P. It is not hard to realize that in fact the action of T on P is without inversions and simply transitive on the edges of P. Consequently, any edge ε of P is a strict fundamental domain for T. Moreover, the stabilizer of the vertex of ε, say u, which is also a vertex of Π/A, is isomorphic to Z_{3}. The stabilizer of the other vertex, say s, is isomorphic to Z_{2}.

Now, define morphism m : F → T by mapping G_{v_{1}} on the stabilizer of s and G_{v_{2}} on the stabilizer of u. Then the development D(F, m) corresponding to this morphism is equivariantly isomorphic to P. It is not hard to see that P is 12-large (in particular, it is also 7-large). We leave it as an exercise to check that P (identified as the development D(F, m)) is extra-tilable. It is the smallest known to us extra-tilable development of F.

We now use the previous graph as one of the local developments for a 2-simplex of groups.

Example 6.5 ((Z_{2}, Z_{2}, Z_{3})-groups). Let Δ be the 2-simplex with vertices v_{0}, v_{1}, v_{2}. Given an integer k ≥ 1, consider the simplex of groups G over Δ with local groups G_{Δ} = {1}, G_{(v_{0},v_{i})} = Z_{2} for i = 1, 2, G_{(v_{1},v_{2})} = Z_{3}, G_{v_{0}} = D_{6k}, G_{v_{i}} = T for i = 1, 2 (where T is the group from Example 6.4), and with structure homomorphisms as in m (see Example 6.4) at vertices v_{1}, v_{2}, and such that ⟨G_{(v_{0},v_{1})}, G_{(v_{0},v_{2})}⟩ = G_{v_{0}}.

The local developments of G at vertices are then isomorphic to a 2k-gon (at v_{0}) and to D(F, m) = P (at v_{1} and v_{2}). In particular, G is locally 7-large and locally extra-tilable. Consequently, in view of Theorem 6.1, the fundamental group π_{1}G is QCS.

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Laboratoire de Mathématiques Université de Paris XI (Paris-Sud), 91405 Orsay, France,

E-mail address: frederic.haglund@math.u-psud.fr

Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

E-mail address: swiatkow@math.uni.wroc.pl